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Question
If f (x) = |3 − x| + (3 + x), where (x) denotes the least integer greater than or equal to x, then f (x) is
Options
continuous and differentiable at x = 3
continuous but not differentiable at x = 3
differentiable nut not continuous at x = 3
neither differentiable nor continuous at x = 3
Solution
(d) neither differentiable nor continuous at x = 3
We have,
\[f\left( x \right) = \left| 3 - x \right| + \left( 3 + x \right), \text { where } \left( x \right) \text{denotes the least integer greater than or equal to} x . \]
`f(x) = {(3-x +3+3,2<x<3),(-3 +x + 3 +4,3<x<4):}`
`⇒ f(x) = {(-x +9,2<x<3),(x+4 , 3<x<4):}`
Here,
\[\left( \text { LHL at x } = 3 \right) = \lim_{x \to 3^-} f\left( x \right) = \lim_{x \to 3^-} \left( - x + 9 \right) = - 3 + 9 = 6\]
\[\left( \text { RHL at x }= 3 \right) = \lim_{x \to 3^+} f\left( x \right) = \lim_{x \to 3^-} \left( x + 4 \right) = 3 + 4 = 7\]
\[\text { Since, } \left( \text { LHL at x } = 3 \right) \neq \left( \text { RHL at x }= 3 \right)\]
\[\text{Hence, given function is not continuous at x} = 3\]
\[\text{Therefore, the function will also not be differentiable at} x = 3\]
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